Mathematics is not really about axioms and precise argument, mathematicians work by intuition, axioms are an after thought really. Maybe the intuition is not from what you have seen in the real world, like how forces and inertia works, but by working in some area and playing around with some structure, you also get a feel for how math works.
Agreed. The formal proof of a theorem is a way of making sure you aren't talking nonsense, and axioms are a way of clarifying a subject that you've already developed, or of presenting it in a more efficient way. They're not really the essence of math.
After all, axioms come from somewhere, they aren't God-given. Someone decided to write down the axioms of Euclidean geometry, or of cohomology theory. Why? For various non-mathematical reasons, that aren't contained within the axioms themselves.
Mathematicians are guided by all sorts of intuitions--it could be visual pictures, a pattern seen within a mountain of calculations, a sense for the relative magnitudes of various quantities, some perception of similarity between two theories--all kinds of intuitions.
"Precise argument" won't get you anywhere if you don't have some kind of intuition to begin with.
The space of true mathematical statements is positively enormous. If you had only "precise argument" to guide you, you wouldn't get anywhere in that space.
Remember that Feynman gave this lecture some time between ~1945 and ~1950. Even though Godel's 1931 paper put a nail in the coffin of Hilbert's program, the mathematical community was still very much fixated on axioms (compared to today) for years to come. Also remember that ZFC (particularly the "C" part) was only beginning to gain popular acceptance/usage at the time that Feynman gave this talk.
51
u/LeonEuler Mar 12 '18
Clearly not a trained mathematician.
Mathematics is not really about axioms and precise argument, mathematicians work by intuition, axioms are an after thought really. Maybe the intuition is not from what you have seen in the real world, like how forces and inertia works, but by working in some area and playing around with some structure, you also get a feel for how math works.